that is the usual braiding isomorphism of Vect on Veven⊗WevenV_{even} \otimes W_{even} and on Veven⊗Wodd⊕Vodd⊗WevenV_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even} but is (−1)(-1) times this on Vodd⊗WoddV_{odd}\otimes W_{odd}.

that is associative and commutative in the usual sense. Specifically for the commutativity this means that with a,b∈Aodda,b \in A_{odd} we have

a⋅b=−b⋅a.
a \cdot b = - b \cdot a
\,.

Whereas if either of aa or bb is in AevenA_{even} we have

a⋅b=b⋅a.
a \cdot b = b \cdot a
\,.

Related notions

Definition

The center of a superalgebra AA is the sub-superalgebra Z(A)↪AZ(A) \hookrightarrow A spanned by all those elements z∈Az \in A of homogeneous degree which graded-commute with all other homogeneois elements aa.

Definition

For AA a superalgebra, its oppositeAopA^{op} is the superalgebra with the same underlying super vector space as AA, and with multiplication defined on homogeneous elements by